1 Calculate the sample mean and standard deviation of the following claim amounts ():

534 671 581 620 401 340 980 845 550 690

[3]

2 Suppose A, B and C are events with 1

2 P(A) = , 1

2 P(B) = , 1

3 P(C) = , 3

4 P(A B) = , 1

6 P(A C) = , 1

6 P(B C) = and 1

12 P(A B C) = . (a) Determine whether or not the events A and B are independent.

(b) Calculate the probability P(A B C). [4]

3 Claim sizes in a certain insurance situation are modelled by a distribution with

moment generating function M(t) given by

M(t) = (1 10t) 2

. Show that E[X

2

] = 600 and find the value of E[X

3

]. [3]

4 Consider a random sample of size 16 taken from a normal distribution with mean

= 25 and variance 2

= 4. Let the sample mean be denoted X . State the distribution of X and hence calculate the probability that X assumes a

value greater than 26. [3]

5 Consider a random sample of size 21 taken from a normal distribution with mean

= 25 and variance 2

= 4. Let the sample variance be denoted S

2

. State the distribution of the statistic 5S

2

and hence find the variance of the statistic S

2

. [3]

6 In a survey conducted by a mail order company a random sample of 200 customers

yielded 172 who indicated that they were highly satisfied with the delivery time of

their orders.

Calculate an approximate 95% confidence interval for the proportion of the

company s customers who are highly satisfied with delivery times. [3]

7 For a group of policies the total number of claims arising in a year is modelled as a

Poisson variable with mean 10. Each claim amount, in units of 100, is independently

modelled as a gamma variable with parameters = 4 and = 1/5.

Calculate the mean and standard deviation of the total claim amount

8 The distribution of claim size under a certain class of policy is modelled as a normal

random variable, and previous years records indicate that the standard deviation is

120.

(i) Calculate the width of a 95% confidence interval for the mean claim size if a

sample of size 100 is available. [2]

(ii) Determine the minimum sample size required to ensure that a 95% confidence

interval for the mean claim size is of width at most 10 . [2]

(iii) Comment briefly on the comparison of the confidence intervals in (i) and (ii)

with respect to widths and sample sizes used. [1]

[Total 5]

9 Let X1

, , Xn

denote a large random sample from a distribution with unknown

population mean and known standard deviation 3. The null hypothesis H0

: = 1 is

to be tested against the alternative hypothesis H1

: > 1, using a test based on the

sample mean with a critical region of the form X k , for a constant k.

It is required that the probability of rejecting H0

when = 0.8 should be

approximately 0.05, and the probability of not rejecting H0

when = 1.2 should be

approximately 0.1.

(i) Show that the test requires

0.8 0.95

3/

k

n

and 1.2 0.10

3/

k

n

where is the standard normal distribution function. [4]

(ii) The values for the sample size n and the critical value k which satisfy the

requirements of part (i) are n = 482 and k = 1.025 (you are not asked to verify

these values).

Calculate the approximate level of significance of the test, and comment on

the value